reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th99:
  -1 <= r & r <= 1 implies 0 <= arccos r & arccos r <= PI
proof
  assume -1 <= r & r <= 1;
  then r in [.-1,1.] by XXREAL_1:1;
  then arccos.r in rng arccos by Th86,FUNCT_1:def 3;
  hence thesis by Th85,XXREAL_1:1;
end;
